Separation principle for nonlinear systems: A bilinear approach (Q2730882)

The authors consider a single-input nonlinear system NEWLINE\[NEWLINE\begin{cases} \dot x=f(x)+ug(x) \\ y=h(x)\end{cases}\tag{1}NEWLINE\]NEWLINE where the vector fields \(f\) and \(g\) are smooth and \(f(0)=g(0)=0\). The map \(h\) is analytic and \(h(0)=0\). Writing \(f(x)= Ax+f_1(x)\), \(g(x)=Bx+ g_1(x)\), \(h(x)=Cx+ h_1(x)\), the authors consider the bilinear system NEWLINE\[NEWLINE\begin{cases} \dot x=Ax+uBx \\ y=Cx \end{cases} \tag{2}NEWLINE\]NEWLINE as an approximating system for (1). Assuming that (2) is observable and stabilizable by a homogeneous (of degree zero) feedback, they construct a state observer for (2). They prove that by using this observer, system (1) can be locally stabilized at the origin. They also prove a separation principle for the case where \(h\) is linear.